Questions tagged [forcing]

Forcing is a set theoretic method used mainly for proving independence results. For questions about forcing function please use (differential-equations).

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1answer
71 views

Mostowski collapse of generic extensions of countable elementary submodels

Let $\lambda$ be a large enough regular cardinal, $\mathbb{P}\in M\prec H_\lambda$, with $M$ countable and $\mathbb{P}$ proper. Let $G$ be $\mathbb P$-generic over the ground model $V$. Let $\pi:M\...
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Intuition for the definition of $M[G]$ in forcing.

I've been going through some basic forcing and although I'm able to mechanically follow the steps I'm lacking any real intuition for it. In particular I'm confused about the evaluations used to ...
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51 views

Weird cofinality behavior without choice?

I vaguely recall the following result, but can't find it or prove it on my own; I'd like to check if it, or something similar to it, is actually true. My memory is that the proof is rather hard, FWIW. ...
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1answer
38 views

Equivalent conditions for $M$-genericity

Suppose $G$ is $M$ generic over some model $M$ of $ZFC$, for some poset $P$. We have the following equivalences: $G$ is $M$-generic, i.e, for every $D \in M$ such that $D$ is dense in $P$, $G \cap D \...
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1answer
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Modified version of generelized Hechler forcing

I'm looking at a modified version of the generalized Hechler forcing: Let $\kappa$ be inaccessible, and fix some $s:\kappa \to \kappa$ strictly monotone taking value in the regular cardinals.* ...
7
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2answers
117 views

What does Cohen independence theorem say?

In his celebrated paper, the independence of the continuum hypothesis, P. Cohen proved that there is a model of Z-F in which the continuum hypothesis fails. As a corollary, continuum hypothesis is ...
7
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1answer
122 views

Preservation of chain conditions

Suppose $\kappa<\lambda$ are regular cardinals, $P$ is a $\kappa$-c.c. poset, and $Q$ is a $\lambda$-c.c. poset. Does forcing with $P$ preserve the $\lambda$-c.c. of $Q$?
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Can we define random forcing through Baire space?

We can talk about Cohen real both in $2^\omega$ and $\omega^\omega$ since all countable forcing notions are equivalent to $\mathbb{C}$, while in random forcing I've only seen random real in $2^\omega$....
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Example of $\kappa$-closed but not $\kappa$-directed closed poset

A poset $(\mathbb{P}, <)$ is $\kappa$-closed iff every descending sequence of length $<\kappa$ has a lower bound. $(\mathbb{P},<)$ is $\kappa$-directed closed iff every directed subset of ...
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38 views

In Cohen forcing $\mathrm{Fn}(\kappa\times\omega, 2)$ prove $(\lambda^\vartheta)^{M[G]} = ((\max\{\lambda, \kappa\})^\vartheta)^M$ [duplicate]

I'm trying to find a solution for exercise G1 of Chapter VII in Kunen's "Set Theory - An Introduction to Independence Proofs". The question is in the context of Cohen forcing. We start with ...
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1answer
81 views

Forcing GCH with iteration of $\mathsf{Add}(\kappa,1)$

For a successor cardinal $\kappa^+$, forcing with $\mathsf{Add}(\kappa^+,1)$ forces $2^\kappa=\kappa^+$ in the extension. Can one use an iteration of this kind of forcing to for the GCH? If so, I'd ...
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1answer
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Adding a real destroys strategic closure

The following should have an elementary argument, but I haven’t been able to think of it. Suppose $P$ is a nontrivial partial order in $V$. Let $r$ be a real not in $V$. Show that in $V[r]$, $P$ is ...
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1answer
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Model theory textbook to learn about the consistency and independency proofs

Is there a model theory textbook that describes how to build boolean-valued models to prove the consistency of a sentence with a theory? It would be great if the book also contained basic techniques ...
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Construction a Bernstein set that is an additive group

I would like to start by showing that there exists a Bernstein set $B$ that is additive group. To see that, it is an easy transfinite induction to construct a Bernstein set that is a linear ...
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1answer
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Product forcing of symmetric systems

Given a family of forcing notions $(P_i)_{i\in I}$ we can take the product $P:=\prod_{i\in I}P_i$ as a forcing notion to create a generic filter of the form $G=(G_i)_{i\in I}$ such that for each $i\in ...
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1answer
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Characterizing Vector Spaces without (provable) Basis in ZF

Without the axiom of choice certain vector spaces cannot be proven to have a (hamel) basis in ZF alone and I am wondering whether there exists some criteria characterizing such spaces. Here is what I ...
5
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1answer
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If $\tau$ is a name, is $\{\sigma:\Vdash \sigma=\tau\}$ a proper class?

Fix some (atomless) poset $\mathbb P$. Let $V^\mathbb{P}$ be the class of $\mathbb P$-names and $\tau\in V^\mathbb{P}$. Consider the collection $$ \{\sigma\in V^\mathbb{P} : \Vdash_\mathbb{P} \sigma=\...
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1answer
53 views

Why does the addition of generic ultra-filters in forcing not lead to inconsistent theories?

Apologies if this is a dumb question - I’m trying to dig into the intricacies of set-theoretic forcing at the moment and am struggling with a step. In particular, given our poset P we need to show ...
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102 views

How to prove Cohen forcing does not add unsplit real?

This is an exercise of a forcing lecture notes. The problem is: Assume $\mathcal{A}$ is an $\omega$-splitting family in $V$. Then $V^\mathbb{C}\models$"$\mathcal{A}$ is $\omega$-splitting". ...
2
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1answer
61 views

Strategically $\kappa$-closed poset that is not $\kappa$-closed.

I'm trying to figure out the difference between being $\kappa$-closed and strategically $\kappa$-closed (in the context of forcing). A poset is $\kappa$-closed if every $\alpha$-chain with $\alpha<\...
5
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1answer
102 views

Rank of element in generic extension versus rank of its name

I sometimes see the following fact used in some arguments: suppose $M[G]$ is a generic extension of $M$ by a forcing $\mathbb P$ and suppose $x\in M[G]$ has rank $<\gamma$, where $\gamma$ is some ...
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1answer
112 views

Is forcing needed to prove existence of a model of ZF that is a countable stage of L?

[Edit]: This is an edit of question [the original question ends before the context is added], it only added the general context of that question, the question itself remains the same. Is forcing ...
4
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1answer
58 views

Ways to construct set models of $\mathsf{ZFC}$ from existing ones

So I'm trying to solve this exercise, but I am having trouble solving it without any further assumptions. This is the exercise: Assume $M$ is a transitive model of $\mathsf{ZFC}$ such that if $\alpha ...
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28 views

Class Forcing Notion and Distributivity

A notion of class forcing $P$ for a model $⟨M,C⟩$ is said to be distributive over $⟨M,C⟩\vDash GB^−$, if for every sequence $⟨D_i| i ∈ I⟩ ∈ C$ of open dense subclasses of P with $i\in M$ and for every ...
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1answer
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Existence of a generic filter $G$ such that $p\in G$ and Axiom of choice

When we prove that "If $M$ is countable, $(P,\le)$ is a partial order and $p\in P$, then there is a $G$ that is $P$-generic over $M$ such that $p\in G$", the axiom of (countable) choice ...
4
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1answer
234 views

An elementary embedding which fixes the ordinals, but is not the identity.

In Woodin's "In the Search of Ultimate-$L$", in section $3.5$ on extenders, he uses an example which I have a hard time understanding. He essentially constructs two transitive models of $\...
5
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1answer
144 views

The failure of product lemma for Sacks forcing

I am reading Jörg Brendle's Bogota note, and the author claimed that Sacks forcing with countable support product does not satisfy a product lemma. Especially, he mentioned that if $\mathbb{S}_I$, $|I|...
5
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1answer
102 views

Models between a ground model and its forcing extension

Jech’s book states and proves a standard result about forcing extensions: Suppose $B$ is a Boolean algebra and let $G$ be $B$-generic over $V$. If $M$ is a model of ZFC such that $V \subset M \subset ...
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1answer
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evaluating gives identity on sets from the basic model

I have a very basic question about forcing and generic set: why for $a\in M$ from the ground model do we have $$\dot{a}[G]=a$$ ? Where it is used in this equality here that $a\in M$ ?
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1answer
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Relativization of a name in forcing

What is the difference between $$\tau \text { is a }\mathbb{P}\text{-name and }\tau\in M$$ and $$(\tau\text{ is a } \mathbb{P}\text{-name})^M?$$ Here $M$ is a ground model. In particular I would like ...
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Ideal textbook to study the indepence of CH

I know that the question about logic, axiomatic set theory and continuum hypothesis (CH) textbooks was asked many times here. Say, the following link contains a huge list of introductory set-theoretic ...
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1answer
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Definition of a rank function related with a dense subset.

Consider $D$ be a filter over $\omega$ and $\mathbb{L}(D)$ be the Laver forcing (i,e, the elements of $\mathbb{L}(D)$ are trees $T\subseteq\omega^{<\omega}$ with stem $s_T$ and $\forall s\in LV_n(T)...
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0answers
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Class forcing and progressively closed iteration

In his published paper ('The Ground Axiom' https://arxiv.org/abs/math/0609064), Jonas Reitz gives the definition of progressively closed iteration (definition 92). Informally, an Ord-length forcing ...
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1answer
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Proper and improper forcing Stationarity

In his book on Proper and Improper forcing Shelah writes on page 89: In Sect. 1 we introduce the property "proper" of forcing notions: preserving stationarity not only of subsets of $\...
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Club set in a finite support iteratiot of c.c.c forcing notions.

I'm reading the Baumgartner and Dordal paper "Adjoining dominating functions" and I have a problem with the Theorem 4.1 proof. My problem is the following: Suppose that $P$ is a c.c.c-...
2
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1answer
78 views

Different definitions of Mathias forcing are equivalent

The Mathias Forcing is defined using increasing sequences and I wondered if it would be forcing equivalent if one omits this condition. So, for a free ultrafilter $U$ define \begin{align*} \mathbb{...
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50 views

Types of Collapsing Forcing Notions

I am interested in different types and properties of ways to make some inaccessible cardinal $\kappa$ the new $\omega_2$. So a way of collapsing every $\alpha<\kappa$ to be of cardinallity $\...
4
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1answer
289 views

Exercise 15.15 of Jech: forcing GCH

Jech uses the following forcing notion to force GCH. For each $\alpha$, let $P_\alpha$ be the notion of forcing that collapses $\beth_{\alpha+1}$ to $\beth_\alpha^+$. $P$ is then the Easton product of ...
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0answers
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Uniformization in the Solovay model

I'm reading section 1.12 in Solovay's A model of Set-Theory in Which Every Set of Reals is Lebesgue Measurable, which aims to prove a uniformization property for definable sets of reals. More ...
5
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1answer
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What common weak choice principles are preserved by forcing?

It is well-known that if you force over a model of $\mathsf{ZFC}$ you get another model of $\mathsf{ZFC}$. Also basic is that the Axiom of Choice itself plays no real part in the mechanics of forcing (...
5
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1answer
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ccc forcing that adds no real which dominates all ground model reals is $\mathfrak{b}$-nondominating

I'm reading Canjar's "Mathias Forcing which does not add dominating reals", where he defines a $\lambda$-cc forcing to be $\lambda$-nondominating if whenever $D$ is a family of reals in $V[G]...
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1answer
105 views

Is there a forcing which only collapses cardinals when squared?

Call a forcing notion $\mathbb{P}$ briefly tame if $\mathbb{P}$ preserves cardinals but $\mathbb{P}^2$ does not. I vaguely recall seeing an argument that such things can't exist, but I can't ...
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3answers
394 views

Why doesn't the fact we can force the continuum hypothesis outright prove the continuum hypothesis?

I'm reading Nick Weaver's Forcing for Mathematicians and in Chapter 12 ("Forcing CH") he starts with this (pg 45 - 46): (Everything here is relativized to $M$ - which in his book is a model ...
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1answer
65 views

Some steps in Jech's presentation of Solovay-Tennenbaum Theorem

I'm working through Theorem 16.13 in Jech, i.e., the consistency of MA with $\neg CH$. I understand the broad idea of the proof, but there are some details that I'm missing. I'm more familiar with the ...
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2answers
138 views

Does forcing leave some trace in the resulting model?

I'm trying to get initiated in the mysteries of forcing, and I decided to try and do so both by 'studying their patent' and by 'reverse engineering', hoping that each method will be of some help to ...
7
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1answer
120 views

Does modal replacement hold in modal logic of forcing?

Let $M$ be a countable model of set theory, and let $\mathbb{M}$ be the generic multiverse on $M$. $\mathbb{M}$ forms a Kripke model for modal logic with $\Diamond \phi$ being true at a world if $\phi$...
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0answers
158 views

Comparing the cardinalities of generic $\mathbb{R}$s

This is yet another question about cardinalities in forcing extensions of models of $\mathsf{ZF+\neg AC}$ (see also here). Specificially, I think I've isolated the simplest question which I can't yet ...
5
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0answers
158 views

Could the “post-forcing continuum” be a new cardinality?

This question is vaguely related to this MO question of mine. Suppose $V$ is a (for simplicity, well-founded) model of $\mathsf{ZF+AD}$. Let $\mathbb{P}$ be a forcing notion in $V$, and let $G$ be $\...
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0answers
38 views

p point: an origin of the acronym

I would like to know what does $P$-point stand for. What it is an abbreviation of in the context of forcing ?
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1answer
116 views

Why is MA not provable from ZFC?

All texts I have seen about Martin's Axiom briefly mention that it is independent of ZFC, that it is implied by CH and that it is relatively consistent with ZFC+(not CH). I have seen proofs of the ...

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